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By C. Jake Williams. May 4, 2010
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First, I'd like to say happy birthday to my mom. She's turning 35, one assumes, and deserves better weather than Utah is providing. I love you mom; see you tonight.
Today I'm finally archiving a spreadsheet I made a few months ago about estimating the value of pi. Download the spreadsheet here.
The basic principle is inscribing one regular polygon inside a circle and inscribing that same circle inside another regular polygon, as shown in this picture.
As the number of sides increases for the polygons their area more closely approximates the area of the circle, regardless of the circle's radius. The number of sides and radius are highlighted in yellow on the spreadsheet, which indicates these two cells can be changed to whatever values you desire.
Remember that you can't have a polygon with fewer than three sides but you can increase without end from there.
As you increase the number of sides, you'll notice that the polygons areas approach the area of the circle. From there, it's easy to calculate an approximation for the value of pi.
To obtain a sufficiently accurate estimate, you need only increase the number of sides to a sufficiently large value.
I invite you to experiment with the spreadsheet as you see fit. You can even unhide the formula cells to make sure you understand the geometry involved! Have fun with it.
This is one way that math was performed before the days of calculators and computers. Thank goodness for progress!
And if you ever need to remember the value of pi, remember the following chant:
You were there.